# Properties

 Label 8280.2.a.bp Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1161328736$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta_1 + 1) q^{7}+O(q^{10})$$ q + q^5 + (b1 + 1) * q^7 $$q + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} + \beta_1 + 2) q^{11} + (\beta_{2} + \beta_1) q^{13} + (\beta_1 + 1) q^{17} + (\beta_{2} - \beta_1 + 2) q^{19} - q^{23} + q^{25} + (\beta_{2} - 1) q^{29} + ( - \beta_{2} - 1) q^{31} + (\beta_1 + 1) q^{35} + ( - \beta_1 + 1) q^{37} + (\beta_{2} + 5) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 6) q^{47} + (\beta_{2} + 4 \beta_1 - 2) q^{49} + (\beta_1 + 1) q^{53} + (\beta_{2} + \beta_1 + 2) q^{55} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{59} + ( - 3 \beta_{2} - \beta_1 + 4) q^{61} + (\beta_{2} + \beta_1) q^{65} + ( - 4 \beta_{2} - \beta_1 + 1) q^{67} + ( - 3 \beta_{2} + 4 \beta_1 + 1) q^{71} + ( - 3 \beta_{2} - 5 \beta_1 + 4) q^{73} + (\beta_{2} + 5 \beta_1 + 4) q^{77} + 4 \beta_{2} q^{79} + ( - 2 \beta_{2} - \beta_1 + 3) q^{83} + (\beta_1 + 1) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{89} + (\beta_{2} + 3 \beta_1 + 2) q^{91} + (\beta_{2} - \beta_1 + 2) q^{95} + (2 \beta_{2} - 2 \beta_1 - 2) q^{97}+O(q^{100})$$ q + q^5 + (b1 + 1) * q^7 + (b2 + b1 + 2) * q^11 + (b2 + b1) * q^13 + (b1 + 1) * q^17 + (b2 - b1 + 2) * q^19 - q^23 + q^25 + (b2 - 1) * q^29 + (-b2 - 1) * q^31 + (b1 + 1) * q^35 + (-b1 + 1) * q^37 + (b2 + 5) * q^41 + (2*b2 + 2*b1) * q^43 + (b2 - b1 + 6) * q^47 + (b2 + 4*b1 - 2) * q^49 + (b1 + 1) * q^53 + (b2 + b1 + 2) * q^55 + (-3*b2 - 2*b1 + 1) * q^59 + (-3*b2 - b1 + 4) * q^61 + (b2 + b1) * q^65 + (-4*b2 - b1 + 1) * q^67 + (-3*b2 + 4*b1 + 1) * q^71 + (-3*b2 - 5*b1 + 4) * q^73 + (b2 + 5*b1 + 4) * q^77 + 4*b2 * q^79 + (-2*b2 - b1 + 3) * q^83 + (b1 + 1) * q^85 + (-2*b2 + 2*b1 - 8) * q^89 + (b2 + 3*b1 + 2) * q^91 + (b2 - b1 + 2) * q^95 + (2*b2 - 2*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} + 4 q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 + 4 * q^7 $$3 q + 3 q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 14 q^{41} + 16 q^{47} - 3 q^{49} + 4 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} + 6 q^{67} + 10 q^{71} + 10 q^{73} + 16 q^{77} - 4 q^{79} + 10 q^{83} + 4 q^{85} - 20 q^{89} + 8 q^{91} + 4 q^{95} - 10 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 + 4 * q^7 + 6 * q^11 + 4 * q^17 + 4 * q^19 - 3 * q^23 + 3 * q^25 - 4 * q^29 - 2 * q^31 + 4 * q^35 + 2 * q^37 + 14 * q^41 + 16 * q^47 - 3 * q^49 + 4 * q^53 + 6 * q^55 + 4 * q^59 + 14 * q^61 + 6 * q^67 + 10 * q^71 + 10 * q^73 + 16 * q^77 - 4 * q^79 + 10 * q^83 + 4 * q^85 - 20 * q^89 + 8 * q^91 + 4 * q^95 - 10 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.76156 −0.363328 3.12489
0 0 0 1.00000 0 −0.761557 0 0 0
1.2 0 0 0 1.00000 0 0.636672 0 0 0
1.3 0 0 0 1.00000 0 4.12489 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8280.2.a.bp yes 3
3.b odd 2 1 8280.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8280.2.a.bk 3 3.b odd 2 1
8280.2.a.bp yes 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{3} - 4T_{7}^{2} - T_{7} + 2$$ T7^3 - 4*T7^2 - T7 + 2 $$T_{11}^{3} - 6T_{11}^{2} + 2T_{11} + 20$$ T11^3 - 6*T11^2 + 2*T11 + 20 $$T_{13}^{3} - 10T_{13} + 8$$ T13^3 - 10*T13 + 8 $$T_{17}^{3} - 4T_{17}^{2} - T_{17} + 2$$ T17^3 - 4*T17^2 - T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 4T^{2} - T + 2$$
$11$ $$T^{3} - 6 T^{2} + 2 T + 20$$
$13$ $$T^{3} - 10T + 8$$
$17$ $$T^{3} - 4T^{2} - T + 2$$
$19$ $$T^{3} - 4 T^{2} - 14 T - 8$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 4 T^{2} - 3 T - 10$$
$31$ $$T^{3} + 2 T^{2} - 7 T - 4$$
$37$ $$T^{3} - 2 T^{2} - 5 T + 8$$
$41$ $$T^{3} - 14 T^{2} + 57 T - 64$$
$43$ $$T^{3} - 40T + 64$$
$47$ $$T^{3} - 16 T^{2} + 66 T - 80$$
$53$ $$T^{3} - 4T^{2} - T + 2$$
$59$ $$T^{3} - 4 T^{2} - 67 T - 142$$
$61$ $$T^{3} - 14 T^{2} - 2 T + 68$$
$67$ $$T^{3} - 6 T^{2} - 109 T - 20$$
$71$ $$T^{3} - 10 T^{2} - 199 T + 1868$$
$73$ $$T^{3} - 10 T^{2} - 130 T + 764$$
$79$ $$T^{3} + 4 T^{2} - 128 T - 256$$
$83$ $$T^{3} - 10 T^{2} + 3 T + 4$$
$89$ $$T^{3} + 20 T^{2} + 56 T + 32$$
$97$ $$T^{3} + 10 T^{2} - 44 T - 472$$